Need help finding a formula for this sequence

Question

A sequence $(x_j)^\infty_{j=0}$ satisfies

$x_1=1$, and for all $m \ge n \ge 0 $ $x_{m+n}+x_{m-n} = \frac12 (x_{2m}+x_{2n})$.

I have to find a formula for $x_j$ and then I can prove that later for homework. Thank you very much.

Answer 1

Try $x_n := n^2$, then we have $(n+m)^2 + (n-m)^2 = 2(n^2 + m^2) = \frac{1}{2}((2m)^2 + (2n)^2)$. Use induction.